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Distributed constrained convex optimization over digraphs: A Fenchel dual based approach

机译：数字上的分布式约束凸优化：基于Fenchel双向方法

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Recently, a lot of distributed optimization algorithms have been established for solving distribution optimization over weight-unbalanced digraphs. Most of these algorithms are designed in discrete-time setting (see, [6], [7], [11] and the references therein). Recently, continuous-time algorithms have attracted considerable attention in solving distributed optimization over weight-unbalanced digraphs. The pioneering work can be dated back to the literature [9], where a continuous-time push-sum protocol is integrated into gradient flow dynamics. Such a mechanism is further applied to saddle-point dynamics in [10] and [12]. A common assumption in [9, 10] and [12] is that the Laplacian matrix of the weight-unbalanced digraph is required to have a zero column sum. However, such a requirement implies that each agent needs to know its out-degree or adjust its outgoing weights as shown in [6, 7]. This is actually unavailable to the agents communicating over a weightunbalanced digraph. In fact, using the Laplacian matrix with a zero row sum in broadcast-based communication networks is realistic. In this case, some results have been reported in [3] and [14]. Actually, for a weight-unbalanced digraph, the continuous-time algorithm in [3] is only able to minimize the weighted sum of local objective functions instead of the sum in the optimization, where the weights are related to the left eigenvalue of Laplacian matrix associated with zero eigenvalue. Such a limitation was avoided in [14] by using an augmented consensus protocol. However, the algorithm in [14] fails to address the case with the intersection of local constraint sets. Here, we consider the distributed convex optimization problem with the intersection of local constraint sets over a strongly connected digraph, where the global objective function is described as a sum of some agents’ local objective functions. To solve the problem in a distributed way, we first resort to its Fenchel dual problem by introducing local conjugate functions. Then, we propose a distributed Fenchel dual gradient algorithm in continuous-time setting via a two-time-scale system. By using the Lyapunov stability theory, we show that under the proposed algorithm, the agents’ decision variables reach consensus and converge asymptotically to the optimal solution when the local objective functions are strongly convex with locally Lipschitz gradients.
机译：最近，已经建立了许多分布式优化算法，用于求解重量不平衡数字的分布优化。这些算法中的大多数都是在离散时间设置（参见，[6]，[7]，[11]和其中的引用）中设计的。最近，连续时间算法在求解重量不平衡数字上的分布式优化方面引起了相当大的关注。开创性工作可以返回到文献[9]，其中连续推送总协议集成到梯度流动动态中。这种机制进一步应用于[10]和[12]中的鞍点动力学。 [9,10]和[12]中的常见假设是重量 - 不平衡数字的拉普拉斯基质需要具有零列和。然而，这种要求意味着每个代理需要了解其Out度或调整其传出权重，如[6,7]所示。这实际上是不可用于在掌控的数字上通信的代理。实际上，使用基于广播的通信网络中的具有零行和的拉普拉斯矩阵是现实的。在这种情况下，[3]和[14]中已报告一些结果。实际上，对于重量不平衡的数字，[3]中的连续时间算法仅能够最小化局部目标函数的加权和，而不是优化中的总和，其中权重与拉普拉斯矩阵的左特征值相关与零特征值相关联。通过使用增强的共识协议在[14]中避免了这种限制。然而，[14]中的算法不能使用本地约束集的交叉来解决这种情况。这里，我们考虑通过强连接的数字上的局部约束集的分布式凸优化问题，其中全局目标函数被描述为一些代理的本地目标函数的总和。要以分布式方式解决问题，我们首先通过引入本地共轭功能来诉诸其Fenchel双问题。然后，我们通过双级系统提出了连续时间设置的分布式Fenchel双梯度算法。通过使用Lyapunov稳定性理论，我们表明，根据所提出的算法，当局部客观函数强烈地凸出的局部Lipschitz梯度时，代理商的决策变量达成共识并收敛到最佳解决方案。

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